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\section{Goods and Service Production}

The production function is:
\begin{equation}
y = Ak
\end{equation}
We assume that the secondary energy, such as electricity, is the only input variable in goods and service production $k = E$. All the effects of other input is included in coefficient $A$. Note that the secondary energy is produced from two different sources: non-renewables $R$ and renewables $B$. So we denote the secondary energy from these two sources seperately as $E_R$ and $E_B$. Then equation $(1)$ changes to:
\begin{equation}
y = A(E_R+E_B)
\end{equation}

\section{Non-renewables $\Rightarrow$ secondary energy}
In this sector, we have two kinds of input: 
\begin{itemize}
\item the nonrenewable energy sources $R$, such as oil, coal and natural gas 
\item the capital $k_R$, such as the equipments in the power plant
\end{itemize}
and one output: $E_R$. The equation is:
\begin{eqnarray}
E_R = \epsilon R = u_R k_R \\
R = \frac{u_R k_R}{\epsilon}
\end{eqnarray}

Where $\epsilon$ is the conversion rate, or energy efficiency rate.

\section{Renewables $\Rightarrow$ secondary energy}
In the renewable energy sector, we also have two input: the primary energy source $B$ and capital $k_B$. The equation is:
\begin{eqnarray}
E_B = B = \frac{k_B}{\mu} 
\end{eqnarray}

\section{Fossil fuel mining}

For the non-renewable energy source, we need do the mining first. The renewable energy doesn't have this step. In this sector, we have the per capita extraction cost $\frac{g(S,N,s)}{Q}$ .

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